Hierarchical interaction can cause turbulent flow even in a weakly nonlinear regime. One of such kinds of turbulence is weak turbulence, which is modeled as spatiotemporal chaos in nonlinear mathematical models. Here, we theoretically study tagged-particle dynamics in a model of weak turbulence, called Nikolaevskii turbulence; especially, the system size dependence is discussed in this paper. The system size does not affect the properties of anomalous diffusion when it is longer than a characteristic length. However, the properties with system size shorter than the characteristic length deviate from those with longer system sizes. It implies that there is a cutoff length in the system size of the Nikolaevskii turbulence. The cutoff length agrees with the critical system size already proposed in the discussion of dynamical exponents. Our results suggest that to investigate the Nikolaevskii turbulence at a low parameter region the system size must be set to be long enough.